Introduction to R

1 R Basics

1.1 Install R and RStudio

1.2 Data type

1.2.1 Matrix & Array

1.2.2 Data frame

1.2.3 List

1.3 Data input and output

1.4 For loop

1.5 If statement

1.6 R function

1.7 Plots

2 Decision tree

2.1 Agenda

• Introduction of Amua
• Creating a decision tree using Amua
• Example
• Exporting Amua decision tree to R script
• Prepare the R decision tree from Amua for further analyses

For a simple decision tree, we can draw the tree easily. As the decision tree grows, there is software helping you draw the tree such as TreeAge and Amua. In this class, we use Amua because it is free. We will show how to use Amua to build a decision tree and export the tree to R script for CEA analysis.

2.2 Install Amua

Follow the instruction here to install Amua. Be aware of the difference between Mac and Windows users! After you install Amua, remember where you store the software.

2.3 Create a decision tree

2.3.1 Open the Amua.jar.

2.3.2 Decision, chance, terminal nodes, and decision tree

2.3.3 Parameterization

2.3.4 Adding more outcomes

2.3.5 Change analysis

2.3.6 Save the model and export the tree to R code

• Notice that Amua created a folder, called temp_Export/.
• In this folder, there are two .R files: main.R and functions.R.
• The body of the decision tree model is in main.R.

2.4 Example

Dracula is preparing for a spring break party tonight. But there’s one final decision that Dracula is struggling with. Being a vampire, he intends to bite and suck the blood of some of his guests at the party (about 25% of them, by his best estimate). While being bitten by a vampire won’t turn his guests into vampires or zombies, like some horror movies might suggest, there is a 50% chance that a vampire bite results in a rather severe bacterial infection, of which 66% of cases require hospitalization for an average of 1 night.

Being a gracious host, Dracula is considering different ways of administering antibiotic prophylaxis to his guests to reduce their risk of infection. One option is to administer the antibiotics to his victim‐guests just before he bites them – this would reduce the risk of a vampire bite infection by 20%. However, the antibiotics can be even more effective, reducing the risk of infection by 90%, if administered at least 30 minutes before being bitten. To achieve this, Dracula is considering putting the antibiotics into the drinks served at the party to ensure that his guests are all properly dosed before he bites his victims. But this means that all his guests would be exposed to the antibiotics (not just those he intends to bite), and he knows that about 5% of people are severely allergic to these antibiotics and would require immediate hospitalization if exposed. Dracula is therefore also considering not administering antibiotic prophylaxis at all to avoid this harm.

However, all the healthy blood doesn’t come without cost. This party will cost Dracula $1000 for a total of 200 guests (an average of $5 per guest). In addition, Dracula expects the cost of antibiotic to be $10 for each guest he bites. If Dracula decides to administer the antibiotics in all the drinks served at the party, the total cost of antibioitics is expected to be $100 (Dracula gets discount for buying a large batch of antibiotics!). Also, Dracula is willing to pay for the cost of hospitalization for any guest who experience bacteiral infection due to his bites because he feels responsible. The cost of hospitalization per person per night is $500. Dracula expects to get an average of 470 mL healthy blood by biting a guest. However, if the guest ends up hospitalized due to bacterial infection, the healthy blood that Dracula can get is reduced by 10%.

Dracula hopes you can help him determine which strategy he should choose.

You can build a decision tree via Amua.

2.5 Wrapper function for decision tree exported from Amua

• The decision tree created in Amua can be exported to R.
• However, the R script created by Amua is not easy to use for advance CEA analysis (e.g., PSA, one/two-way sensitivity analysis, EVPI, etc.).
• We created some wrapper functions to reshape the R script from Amua. This transformation of code allows us to do further analysis using other packages in R.
• Install the package using the following code.

if(!require("remotes")) install.packages("remotes")
if(!require("CEAutil")) remotes::install_github("syzoekao/CEAutil", dependencies = TRUE)

library(CEAutil)

• We will use two functions from the package to convert the R script exported from Amua.

1. parse_amua_tree():

This function only takes an input argument, the path to the main.R file of the Amua decision model. The function returns a list of outputs. Output 1 param_ls is a list of input parameters with basecase values used in Amua. Output 2 treefunc is the R code of the Amua decision model formatted as text.

treetxt <- parse_amua_tree("AmuaExample/DraculaParty_Export/main.R")

print(treetxt$param_ls)

## $p_inf_bitten
## [1] 0.5
## 
## $p_inf_not_allergic
## [1] 0.4
## 
## $p_inf_bitten_UA
## [1] 0.05
## 
## $C_hospital
## [1] 500
## 
## $C_donothing
## [1] 5
## 
## $C_drug
## [1] 10
## 
## $C_drug_UA
## [1] 0.5
## 
## $p_hospital
## [1] 0.66
## 
## $p_bite
## [1] 0.25
## 
## $p_not_allerg
## [1] 0.95
## 
## $red_blood
## [1] 0.1

2. dectree_wrapper():

We use the two outputs from the parse_amua_tree() as the input arguments in the dectree_wrapper(). The input arguments in the wrapper function include params_basecase, treefunc, and popsize. params_basecase takes a list of named input parameters. treefunc takes the text file organized by the parse_amua_tree(). popsize is defaulted as 1 but you could change your population size.

param_ls <- treetxt[["param_ls"]]
treefunc <- treetxt[["treefunc"]]

tree_output <- dectree_wrapper(params_basecase = param_ls, treefunc = treefunc, popsize = 200)

• The cost and effectiveness of each of Dracula’s strategy based on the basecase values

print(tree_output)

##                   name expectedEff1 expectedEff2    Cost
## 1            Donothing     10974.50      16.5000 9250.00
## 2 Universalantibiotics     22251.33      11.5675 6883.75
## 3  Targetedantibiotics     21735.62      15.0400 9020.00

2.6 Change parameter values

• Many of the parameters are organized as a variable in the decision tree. We can easily see how the cost and effectiveness vary with changes of parameters of interest.

Example 1. Dracula has been starved over the long cruel winter in Minnesota. The Spring break is the first time that his guests are willing to come to his party in several months. Therefore, Dracula is going to seize the chance to bite as many guests as possible. The probability that he bites a guest is now increased to 50%. What are the cost and effectiveness of each strategy?

param_ls$p_bite <- 0.5

tree_output <- dectree_wrapper(params_basecase = param_ls, treefunc = treefunc, popsize = 200)

print(tree_output)

##                   name expectedEff1 expectedEff2    Cost
## 1            Donothing     21949.00       33.000 17500.0
## 2 Universalantibiotics     44502.65       13.135  7667.5
## 3  Targetedantibiotics     43471.24       30.080 17040.0

Example 2. The cost of hospitalization due to bacterial infection varies from guest to guest depending on the healthcare that a guest has. Overall, the cost of hospitalization has a mean of $500 with a standard deviation $300 following a gamma distribution. What are the mean cost and effectiveness of the party across all 200 guests?

if(!require(dampack)) remotes::install_github("DARTH-git/dampack", dependencies = TRUE)

require(dampack)

C_hospital <- gen_psa_samp(params = c("C_hospital"), 
                           dists = c("gamma"), 
                           parameterization_types = c("mean, sd"), 
                           dists_params = list(c(500, 250)), 
                           nsamp = 200)

cat("mean and sd are", c(mean(C_hospital$C_hospital), sd(C_hospital$C_hospital)), ", respectively.")

## mean and sd are 476.7693 232.8625 , respectively.

hist(C_hospital$C_hospital, main = "histogram", xlab = "values", ylab = "frequency")

tree_vary <- dectree_wrapper(params_basecase = param_ls, 
                       treefunc = treefunc, 
                       popsize = 1, 
                       vary_param_samp = C_hospital)

• The elements in the tree_vary output

print(names(tree_vary))

## [1] "expectedEff1" "expectedEff2" "Cost"         "param_samp"

• Let’s take a look at some elements in the tree_vary output

print(head(tree_vary$param_samp))

##   nsamp C_hospital
## 1     1   688.8988
## 2     2   183.3789
## 3     3   311.3923
## 4     4   293.4517
## 5     5   250.6182
## 6     6   603.3601

print(head(tree_vary$expectedEff1))

##   Donothing Universalantibiotics Targetedantibiotics
## 1   109.745             222.5133            217.3562
## 2   109.745             222.5133            217.3562
## 3   109.745             222.5133            217.3562
## 4   109.745             222.5133            217.3562
## 5   109.745             222.5133            217.3562
## 6   109.745             222.5133            217.3562

print(head(tree_vary$expectedCost))

## NULL

• The mean cost

print(summary(tree_vary$expectedCost))

## Length  Class   Mode 
##      0   NULL   NULL

3 Markov model

3.1 Agenda

• General Markov model coding structure
• Follow the coding steps using an example
• The template of Markov model function
• Consideration of strategies
• Prepare your Markov model for further analyses

3.2 General Markov model coding structure

• We will show how to code a Markov model step by step.
• In general, we follow this structure:

markov_model <- function(l_param_all,
                         strategy = NULL) {
  #### 1. Read in, define, or transform parameters if needed
  
  #### 2. Create the transition probability matrices using array
  
  #### 3. Create the trace matrix to track the changes in the population distribution 
  #### through time. You could also create other matrix to track different outcomes,
  #### e.g., costs, incidence, etc.
  
  #### 4. Get outputs
  
  #### 5. Return the relevant results
}

• Then we write this process into a function markov_model()

3.3 Example

The Canadian province of Ontario is considering a province-wide ban on indoor tanning as a means of preventing skin cancer, with a focus on young women. The Ontario Ministry of Health has just finished a large observational study on tanning behaviors and skin cancer incidence among women in Ontario to inform their decision.

In their surveillance study, they found that skin cancer risks differ substantially for individuals who visit tanning salons regularly (“regular tanners”) and those who do not (“non-tanners”). The annual risk of skin cancer was 4% for regular tanners vs. 0.5% for non-tanners. (Assume that skin cancer risk depends only on current tanning behavior and not on tanning history).

The Ministry of Health also studied tanning behaviors. They estimated the annual probability of a non-tanner becoming a regular tanner, by age, among women. This probability begins increasing around age 12, peaks at age 24 and then begins to decrease. They also estimated the rate of a regular tanner becoming a non-tanner. This probability is low until age 30, after which it increases with age. These data are summarized in the dataset ONtan in the CEAutil package.

Skin cancer resolves within one year of diagnosis, with 7% of cases resulting in death. Those who recover following a skin cancer diagnosis nearly all quit tanning, though they re-initiate indoor tanning at the same rate as their peers who have not experienced skin cancer.

The Ontario Ministry of Health is considering an indoor tanning ban for those 18 years of age and younger (reducing the rate of tanning initiation to zero among this age group). They are also considering a full indoor tanning ban, which would reduce the rate of tanning initiation to zero for everyone in Ontario. They would like to evaluate the impact that each of these policies would have over the lifetime of a cohort of 10-year-old girls. The Ministry of Health does not wish to discount outcomes.

1. Draw a Markov model diagram of tanning behavior and skin cancer that the Ontario Ministry of Health could use to evaluate their tanning policies in terms of the desired outcomes. Use a yearly time step. Include the following states: “Non-tanners”, “Regular tanners”, “Skin cancer”, “Dead from skin cancer”, and “Dead from other causes”. Label all transition probabilities and indicate which are changing over time.

2. Calcuate the remaining life-expectancy of a 10-year-old girl under each strategy.

3.3.1 Step 0. Model setup

Before doing any actual coding:

1. Draw your Markov model diagram
2. Figure out your parameters

• which are fixed?
• which are time varying?
• do you have parameters that require calibration?

3. What is the time horizon?

• 100 years

4. What is the cycle length?

• annual time step

Here is the Markov model diagram:

3.3.2 Step 1. Set up parameters, data, and variables required

• We start with setting up parameter values, function, or transformation.
• The parameters included in this example:

parameter code	definition	data type
p_init_tan	probability of initiating tanning	table
p_halt_tan	probability of discontinuing tanning	table
p_nontan_to_cancer	probability of having skin cancer among non-tanners	constant
p_regtan_to_cancer	probability of having skin cancer among regular tanners	constant
p_cancer_to_dead	probability of cancer-specific death	constant
p_mort	probability of natural death	table

• You could find the tanning behavior and the life table in the CEAutil package.

library(CEAutil)

data(ONtan)
ltable <- ONtan$lifetable
behavior <- ONtan$behavior

print(head(ltable))

##   age      qx
## 1   0 0.00468
## 2   1 0.00017
## 3   2 0.00014
## 4   3 0.00012
## 5   4 0.00011
## 6   5 0.00009

print(head(behavior))

##   age p_init_tanning p_halt_tanning
## 1  10           0.05           0.01
## 2  11           0.05           0.01
## 3  12           0.10           0.01
## 4  13           0.10           0.01
## 5  14           0.11           0.01
## 6  15           0.12           0.01

• For the constant parameters:

p_nontan_to_cancer <- 0.005
p_regtan_to_cancer <- 0.04
p_cancer_to_dead <- 0.07

• In addition, we have other required variables
• timehorizon (n_t)
• names of each health state (state_names)
• initial status (v_init)

n_t <- 100 
state_names <- c("nontan", "regtan", "cancer", "deadnature", "deadcancer")
v_init <- c(1, 0, 0, 0, 0)

• These input parameters are often defined outside the markov_model() function. In this model framework, we use list() to combine all the parameters, data, and variables defined beforehand. We provide the name for each the element in the list.

l_param_all <- list(p_nontan_to_cancer = 0.005,
                    p_regtan_to_cancer = 0.04,
                    p_cancer_to_dead = 0.07, 
                    ltable = ltable,
                    behavior = behavior, 
                    state_names = c("nontan", "regtan", "cancer", "deadnature", "deadcancer"),
                    n_t = 100,
                    v_init = c(1, 0, 0, 0, 0))

• In the function, we start with some basic calculation and transformation of the input parameters, data, and variables that will be used in the Markov model.

#### 1. Read in, set, or transform parameters if needed
# We need the age index for matching values of the lifetable and behavior data.
ages <- c(10 : (10 + n_t - 1))

# We extract the probability of natural death age 10-110 from the lifetable. 
p_mort <- ltable$qx[match(ages, ltable$age)]

# We modify the values of tanning behavior based on strategy of interest.
if (strategy == "targeted ban") {
  behavior$p_init_tanning[behavior$age <= 18] <- 0
}
if (strategy == "universal ban") {
  behavior$p_init_tanning <- 0
}

# We extract the tanning behavior at age 10-110 from the behavior data 
p_init_tan <- behavior$p_init_tanning[match(ages, behavior$age)]
p_halt_tan <- behavior$p_halt_tanning[match(ages, behavior$age)]

# We get the number of health states based on the length of the string vector, state_names
n_states <- length(state_names)

3.3.3 Step 2. Create the transition probability array

• A transition probability matrix contains the transition probability from one health state to another state.

##        state1 state2 state3
## state1    0.1    0.2    0.7
## state2    0.5    0.1    0.4
## state3    0.7    0.0    0.3

• In many of our applications, individuals transit from the row state to the column state. Each row should sum up to 1.

print(rowSums(transit_matrix))

## state1 state2 state3 
##      1      1      1

• The transition probability array can contain more than one transition matrix. Because the behavior and mortality vary over time. The transition matrix is different from year to year. Thus, we create a transition array with the following dimension \(n\_state \times n\_state \times n\_t\).

tr_mat <- array(0, dim = c(n_states, n_states, n_t),
                dimnames = list(state_names, state_names, ages))

• We initiate the transition array with 0 because most transition probabilities are zeros.
• Then we modify the transition probabilities for each origin state.

# 1. Fill out the transition probabilities from non-tanner to other states
tr_mat["nontan", "regtan", ] <- p_init_tan
tr_mat["nontan", "cancer", ] <- p_nontan_to_cancer
tr_mat["nontan", "deadnature", ] <- p_mort
tr_mat["nontan", "nontan", ] <- 1 - p_init_tan - p_nontan_to_cancer - p_mort

# 2. Fill out the transition probabilities from regular tanner to other states
tr_mat["regtan", "nontan", ] <- p_halt_tan
tr_mat["regtan", "cancer", ] <- p_regtan_to_cancer
tr_mat["regtan", "deadnature", ] <- p_mort
tr_mat["regtan", "regtan", ] <- 1 - p_halt_tan - p_regtan_to_cancer - p_mort

# 3. Fill out the transition probabilities from skin cancer to other states.
#    Be careful that this is a tunnel state. Therefore, there is no self loop.
tr_mat["cancer", "deadcancer", ] <- p_cancer_to_dead
tr_mat["cancer", "deadnature", ] <- p_mort
tr_mat["cancer", "nontan", ] <- 1 - p_cancer_to_dead - p_mort

# 4. Fill out the transition probabilities for cancer specific death (this is an absorbing state!!)
tr_mat["deadcancer", "deadcancer", ] <- 1

# 5. Fill out the transition probabilities for natural death (this is an absorbing state!!)
tr_mat["deadnature", "deadnature", ] <- 1

• Let’s see two slice of the transition array

print(tr_mat[, , "20"])

##             nontan  regtan cancer deadnature deadcancer
## nontan     0.82477 0.17000  0.005    0.00023       0.00
## regtan     0.01000 0.94977  0.040    0.00023       0.00
## cancer     0.92977 0.00000  0.000    0.00023       0.07
## deadnature 0.00000 0.00000  0.000    1.00000       0.00
## deadcancer 0.00000 0.00000  0.000    0.00000       1.00

print(tr_mat[, , "50"])

##             nontan  regtan cancer deadnature deadcancer
## nontan     0.98303 0.01000  0.005    0.00197       0.00
## regtan     0.50000 0.45803  0.040    0.00197       0.00
## cancer     0.92803 0.00000  0.000    0.00197       0.07
## deadnature 0.00000 0.00000  0.000    1.00000       0.00
## deadcancer 0.00000 0.00000  0.000    0.00000       1.00

• Check whether the transition array meets the criteria of transition matrix

# Check whether the transition matrices have any negative values or values > 1!!!
if (sum(tr_mat > 1) | sum(tr_mat < 0)) stop("there are invalid transition probabilities")

# Check whether each row of a transition matrix sum up to 1!!!
if (any(rowSums(t(apply(tr_mat, 3, rowSums))) != n_states)) stop("transition probabilities do not sum up to one")

3.3.4 Step 3. Create the trace matrix and outcome matrix

• Initialize the trace matrix and replace the first row with the initial state (v_init)

#### 3. Create the trace matrix to track the changes in the population distribution through time
#### You could also create other matrix to track different outcomes,
#### e.g., costs, incidence, etc.
trace_mat <- matrix(0, ncol = n_states, nrow = n_t + 1,
                    dimnames = list(c(10 : (10 + n_t)), state_names))
# Modify the first row of the trace_mat using the v_init
trace_mat[1, ] <- v_init

• If you are interested in tracking other outcomes, initialize an empty matrix here.

# Suppose that we want to track the cost of having skin cancer for a year
trace_cost <- rep(0, n_t)

3.3.5 Step 4. Compute the Markov model over time

• We use for() loop to iterate the calculation through time.

#### 4. Compute the Markov model over time by iterating through time steps
for(t in 1 : n_t){
  trace_mat[t + 1, ] <- trace_mat[t, ] %*% tr_mat[, , t]
}

3.3.6 Step 5. Organize outputs

• The outcome in this example is “expected life years” (LE). Therefore, we remove the last two columns.
• Calculate the sums by each row for the remaining columns.

• Calculate the life expectancy by summing up all the values.

• We organize the output in a data.frame with the strategy in the first column and LE as the second column.

• You could add more columns to the data.frame if you have more outcomes of interest (e.g., cost)

#### 5. Organize outputs
LE <- sum(rowSums(trace_mat[, -c(4, 5)])) - 1
output <- data.frame("strategy" = "null",
                     "LE" = LE)

3.3.7 Step 6. Return the relevant outputs

• Finally, return the outputs/results from the function.

return(output)

• You could return a list of outputs. For example, if you want to also return the trace matrix, you could do this:

return(list(output = output, trace = trace_mat))

3.4 The template Markov model function

• You could find the template Markov model function in the CEAutil package.
• You can modify the function for your project.

Need discussion!! Should we illustrate how to calculate cost?

library(CEAutil)

print(markov_model)

## function(l_param_all,
##                          strategy = NULL) {
##   with(as.list(l_param_all), {
##     #### Step 1. Read in, set, or transform parameters if needed
##     ages <- c(10 : (10 + n_t - 1))
##     p_mort <- ltable$qx[match(ages, ltable$age)]
## 
##     if (strategy == "targeted ban") {
##       behavior$p_init_tanning[behavior$age <= 18] <- 0
##     }
##     if (strategy == "universal ban") {
##       behavior$p_init_tanning <- 0
##     }
## 
##     p_init_tan <- behavior$p_init_tanning[match(ages, behavior$age)]
##     p_halt_tan <- behavior$p_halt_tanning[match(ages, behavior$age)]
## 
##     n_states <- length(state_names)
## 
##     #### Step 2. Create the transition probability matrices using array
##     tr_mat <- array(0, dim = c(n_states, n_states, n_t),
##                     dimnames = list(state_names, state_names, ages))
## 
##     ## Start filling out transition probabilities in the array
##     ## When you fill out the transition probabilities, always remember to deal with
##     ## the self-loop the LAST!!!!!
##     # 1. Fill out the transition probabilities from non-tanner to other states
##     tr_mat["nontan", "regtan", ] <- p_init_tan
##     tr_mat["nontan", "cancer", ] <- p_nontan_to_cancer
##     tr_mat["nontan", "deadnature", ] <- p_mort
##     tr_mat["nontan", "nontan", ] <- 1 - p_init_tan - p_nontan_to_cancer - p_mort
## 
##     # 2. Fill out the transition probabilities from regular tanner to other states
##     tr_mat["regtan", "nontan", ] <- p_halt_tan
##     tr_mat["regtan", "cancer", ] <- p_regtan_to_cancer
##     tr_mat["regtan", "deadnature", ] <- p_mort
##     tr_mat["regtan", "regtan", ] <- 1 - p_halt_tan - p_regtan_to_cancer - p_mort
## 
##     # 3. Fill out the transition probabilities from skin cancer to other states.
##     #    Be careful that this is a tunnel state. Therefore, there is no self loop.
##     tr_mat["cancer", "deadcancer", ] <- p_cancer_to_dead
##     tr_mat["cancer", "deadnature", ] <- p_mort
##     tr_mat["cancer", "nontan", ] <- 1 - p_cancer_to_dead - p_mort
## 
##     # 4. Fill out the transition probabilities for cancer specific death (this is an absorbing state!!)
##     tr_mat["deadcancer", "deadcancer", ] <- 1
## 
##     # 5. Fill out the transition probabilities for natural death (this is an absorbing state!!)
##     tr_mat["deadnature", "deadnature", ] <- 1
## 
##     # Check whether the transition matrices have any negative values or values > 1!!!
##     if (sum(tr_mat > 1) | sum(tr_mat < 0)) stop("there are invalid transition probabilities")
## 
##     # Check whether each row of a transition matrix sum up to 1!!!
##     if (any(rowSums(t(apply(tr_mat, 3, rowSums))) != n_states)) stop("transition probabilities do not sum up to one")
## 
##     #### Step 3. Create the trace matrix to track the changes in the population distribution through time
##     #### You could also create other matrix to track different outcomes,
##     #### e.g., costs, incidence, etc.
##     trace_mat <- matrix(0, ncol = n_states, nrow = n_t + 1,
##                         dimnames = list(c(10 : (10 + n_t)), state_names))
##     # Modify the first row of the trace_mat using the v_init
##     trace_mat[1, ] <- v_init
## 
##     # Suppose that we want to track the cost of having skin cancer for a year
##     trace_other <- matrix(0, ncol = 2, nrow = n_t + 1,
##                           dimnames = list(c(10 : (10 + n_t))))
## 
##     #### Step 4. Compute the Markov model over time by iterating through time steps
##     for(t in 1 : n_t){
##       trace_mat[t + 1, ] <- trace_mat[t, ] %*% tr_mat[, , t]
##     }
## 
##     #### Step 5. Organize outputs
##     LE <- sum(rowSums(trace_mat[, !grepl("dead", state_names)])) - 1
##     output <- data.frame("strategy" = strategy,
##                          "LE" = LE)
## 
##     #### Step 6. Return the relevant results
##     return(output)
##   })
## }
## <bytecode: 0x7fcd5c172038>
## <environment: namespace:CEAutil>

3.5 Strategies

• Notice that the markov_model() function has another argument strategy.
• Specifying different strategy produces different life expectancy.
• In our example, there are three types of strategies: “null”, “targeted ban”, “universal ban”. The default value is NULL, which means null strategy.

print(markov_model(l_param_all = l_param_all, strategy = "null"))

##   strategy       LE
## 1     null 70.55568

print(markov_model(l_param_all = l_param_all, strategy = "targeted ban"))

##       strategy       LE
## 1 targeted ban 71.24541

print(markov_model(l_param_all = l_param_all, strategy = "universal ban"))

##        strategy       LE
## 1 universal ban 72.44861

• In our example, the strategy is modifying the probability of initiating tanning. Therefore, we program the effect of strategy at step 1.
• However, strategies are sometimes more complicated to capture. The influence of different strategy might occur in step 2 and step 4 (if the calculation of transition probability at step t depends on the state of step t - 1).

3.6 Markov model wrapper function

• Calculating the results for each strategy one by one is easy for one set of parameters but might become more and more tedious when the scope of the job is growing.
  
• markov_decision_wrapper() is to calculate the results of all strategies all at once.
• vary_param_ls: the parameters that could change in other more advance analyses
• other_input_ls: the paramters, datasets, and variables that do not change from simulation to simulation
• userfun: the Markov model function. You could also create your own Markov model function
• strategy_set: The vector of the strategy names.

vary_param_ls <- list(p_nontan_to_cancer = 0.005,
                      p_regtan_to_cancer = 0.04,
                      p_cancer_to_dead = 0.07)

other_input_ls <- list(ltable = ltable,
                       behavior = behavior,
                       state_names = c("nontan", "regtan", "cancer", "deadnature", "deadcancer"),
                       n_t = 100,
                       v_init = c(1, 0, 0, 0, 0))
res <- markov_decision_wrapper(vary_param_ls = vary_param_ls,
                               other_input_ls = other_input_ls,
                               userfun = markov_model,
                               strategy_set = c("null", "targeted ban", "universal ban"))
print(res)

##        strategy       LE
## 1          null 70.55568
## 2  targeted ban 71.24541
## 3 universal ban 72.44861

• We can change the values in the l_param_all easily.

vary_param_ls <- list(p_nontan_to_cancer = 0.005,
                      p_regtan_to_cancer = 0.08,
                      p_cancer_to_dead = 0.1)

res <- markov_decision_wrapper(vary_param_ls = vary_param_ls,
                               other_input_ls = other_input_ls,
                               userfun = markov_model,
                               strategy_set = c("null", "targeted ban", "universal ban"))
print(res)

##        strategy       LE
## 1          null 67.24277
## 2  targeted ban 68.95024
## 3 universal ban 72.04560

4 dampack

4.1 Agenda

• General introduction of dampack
• One-way sensitivity analysis
• Two-way sensitivity analysis
• Probabilistic sensitivity analysis

4.2 Introduction of dampack

• dampack is the decision analysis modeling package here.
• We have used some functionality in the past.
  
• This package includes functionalities for basic and advanced decision analysis.
• We will illustrate deterministic analysis and probabilistic analysis using the Dracula example.

library(CEAutil)
library(dampack)

4.3 One-way sensitivity analysis

• Investigate the changes in the outcomes by varying one parameter at a time.
• Dracula is interested in how the cost and effectiveness change with probabililty of biting a guest (p_bite), the cost of the antibiotics (C_drug), and the probability of infection after he bites a guest (p_inf_bitten), respectively
• run_owsa_det() is for the one-way sensitivity analysis. Let’s take a look into the run_owsa_det() function:

run_owsa_det(params_range, params_basecase, nsamp = 100, FUN, 
             outcomes = NULL, strategies = NULL, ...)

• params_range: a data.frame with 3 columns in the following order: “pars”, “min”, and “max”.

params_range <- data.frame(pars = c("p_bite", "C_drug", "p_inf_bitten"), 
                           min = c(0.05, 2, 0.2), 
                           max = c(0.8, 20, 0.9))
print(params_range)

##           pars  min  max
## 1       p_bite 0.05  0.8
## 2       C_drug 2.00 20.0
## 3 p_inf_bitten 0.20  0.9

• params_basecase: a named list of basecase values for input parameters needed by the user-defined decision function FUN.

treetxt <- parse_amua_tree("AmuaExample/DraculaParty_Export/main.R")
treefunc <- treetxt[["treefunc"]] # This is the tree function text from Amua
param_ls <- treetxt[["param_ls"]] # This is a list of parameters with basecase values
print(param_ls)

## $p_inf_bitten
## [1] 0.5
## 
## $p_inf_not_allergic
## [1] 0.4
## 
## $p_inf_bitten_UA
## [1] 0.05
## 
## $C_hospital
## [1] 500
## 
## $C_donothing
## [1] 5
## 
## $C_drug
## [1] 10
## 
## $C_drug_UA
## [1] 0.5
## 
## $p_hospital
## [1] 0.66
## 
## $p_bite
## [1] 0.25
## 
## $p_not_allerg
## [1] 0.95
## 
## $red_blood
## [1] 0.1

• nsamp: number of samples. Default is 100.

• FUN: user-defined function that takes the basecase in params_basecase and ... to produce the outcome(s) of interest. The FUN must return a dataframe where the first column are the strategy names and the rest of the columns must be outcomes. The function used in our tree example is dectree_wrapper().

• outcomes: The outcomes of interest. If you use the default setting NULL, run_owsa_det will run the one-way sensitivity analysis for every outcome. In our case, the outcomes include expectedEff1, expectedEff2, and Cost. You could also specify a subset of outcomes:

outcomes = c("expectedEff1", "Cost")

• strategies: You can leave this as default or give your favorite names to the strategies.

• ...: Additional arguments for the user-defined function FUN. In our case, dectree_wrapper() requires params_basecase, treefunc, and popsize. We have already set up the params_basecase. Thus, for ..., we need to add treefunc and popsize.

owsa_out <- run_owsa_det(params_range, param_ls, nsamp = 100, 
                         dectree_wrapper, 
                         treefunc = treefunc, popsize = 200)
print(str(owsa_out))

## List of 3
##  $ owsa_expectedEff1:Classes 'owsa' and 'data.frame':    900 obs. of  4 variables:
##   ..$ parameter  : Factor w/ 3 levels "p_bite","C_drug",..: 1 1 1 1 1 1 1 1 1 1 ...
##   ..$ strategy   : Factor w/ 3 levels "st_Donothing",..: 1 1 1 1 1 1 1 1 1 1 ...
##   ..$ param_val  : num [1:900] 0.05 0.0576 0.0652 0.0727 0.0803 ...
##   ..$ outcome_val: num [1:900] 2195 2527 2860 3193 3525 ...
##  $ owsa_expectedEff2:Classes 'owsa' and 'data.frame':    900 obs. of  4 variables:
##   ..$ parameter  : Factor w/ 3 levels "p_bite","C_drug",..: 1 1 1 1 1 1 1 1 1 1 ...
##   ..$ strategy   : Factor w/ 3 levels "st_Donothing",..: 1 1 1 1 1 1 1 1 1 1 ...
##   ..$ param_val  : num [1:900] 0.05 0.0576 0.0652 0.0727 0.0803 ...
##   ..$ outcome_val: num [1:900] 3.3 3.8 4.3 4.8 5.3 5.8 6.3 6.8 7.3 7.8 ...
##  $ owsa_Cost        :Classes 'owsa' and 'data.frame':    900 obs. of  4 variables:
##   ..$ parameter  : Factor w/ 3 levels "p_bite","C_drug",..: 1 1 1 1 1 1 1 1 1 1 ...
##   ..$ strategy   : Factor w/ 3 levels "st_Donothing",..: 1 1 1 1 1 1 1 1 1 1 ...
##   ..$ param_val  : num [1:900] 0.05 0.0576 0.0652 0.0727 0.0803 ...
##   ..$ outcome_val: num [1:900] 2650 2900 3150 3400 3650 3900 4150 4400 4650 4900 ...
## NULL

print(head(owsa_out$owsa_Cost))

##   parameter     strategy  param_val outcome_val
## 1    p_bite st_Donothing 0.05000000        2650
## 2    p_bite st_Donothing 0.05757576        2900
## 3    p_bite st_Donothing 0.06515152        3150
## 4    p_bite st_Donothing 0.07272727        3400
## 5    p_bite st_Donothing 0.08030303        3650
## 6    p_bite st_Donothing 0.08787879        3900

• We can see how each parameter affects the cost and effectiveness.

plot(owsa_out$owsa_expectedEff1)

plot(owsa_out$owsa_Cost)

4.4 Two-way sensitivity analysis

• Investigate the changes in the outcomes by varying two parameters simultaneously.
• Dracula wants to know how the cost and effectiveness changes as p_bite and C_drug both changes.
• run_twsa_det() allows us to investigate two-way sensitivity analysis for any pair of parameters.

run_twsa_det(params_range, params_basecase, nsamp = 40,
  FUN, outcomes = NULL, strategies = NULL, ...)

• The structure of the function is very similar to run_owsa_det(). The primary difference is the function can only take two parameters at a time in the params_range.

params_range <- data.frame(pars = c("p_bite", "C_drug"), 
                           min = c(0.05, 2), 
                           max = c(0.8, 20))

• Run the two-way sensitivity analysis and see the two-way plots.

twsa_out <- run_twsa_det(params_range, param_ls, nsamp = 10, 
                         dectree_wrapper, 
                         treefunc = treefunc, popsize = 200)
plot(twsa_out$twsa_expectedEff1)

plot(twsa_out$twsa_Cost)

4.5 Probabilistic sensitivity analysis

Need discussion!!

4.5.1 Generate your PSA samples

• gen_psa_samp()

4.5.2 Conduct PSA

• run_psa